Rapid determination of solid-state diffusion coefficients in Li-based batteries via intermittent current interruption method

The galvanostatic intermittent titration technique (GITT) is considered the go-to method for determining the Li+ diffusion coefficients in insertion electrode materials. However, GITT-based methods are either time-consuming, prone to analysis pitfalls or require sophisticated interpretation models. Here, we propose the intermittent current interruption (ICI) method as a reliable, accurate and faster alternative to GITT-based methods. Using Fick’s laws, we prove that the ICI method renders the same information as the GITT within a certain duration of time since the current interruption. Via experimental measurements, we also demonstrate that the results from ICI and GITT methods match where the assumption of semi-infinite diffusion applies. Moreover, the benefit of the non-disruptive ICI method to operando materials characterization is exhibited by correlating the continuously monitored diffusion coefficient of Li+ in a LiNi0.8Mn0.1Co0.1O2-based electrode to its structural changes captured by operando X-ray diffraction measurements.


Supplementary Figure 8. Examples of GITT and ICI measurements
The electrode potential (E) is plotted against the step time (t) in the square root scale for all subplots. In the two examples of GITT measurements a below 3.7 V and b above 3.7 V, results from linear regression of potential against the square root of step time with datapoints in 5-40 s and 50-150 s intervals are also plotted as red and orange lines, respectively. c is an example of an ICI measurement, which took place after the GITT measurement in b. The red line shows the result from linear regression of potential against the square root of step time with datapoints in 1-5 s interval. The relative difference between the k values from the GITT and ICI methods (kGITT and kICI, respectively) above 3.7 V in Figure 3 in the main text (Cell1 in cycle 1) plotted against the OCP of the electrode (E) against Li/Li + . The average, 0.013, is shown by the horizontal line and the standard deviation, 0.26, is shown by the height of the shaded area above and below the average value.
Supplementary Figure 10. Selected impedance spectra from the first discharge of Cell 1. The impedance spectra measured when the working electrode potential is above and below 3.7 V are presented in Nyquist plots in panels a and b, respectively. Z' and Z'' are the real and imaginary parts of the impedance.  Figure 11. Statistical analysis on the relative difference between the slopes of OCP from the GITT and ICI methods. The relative difference between the slope of the OCP (denoted as dEOC/dtI in Figure 4 in the main text) from the GITT and ICI methods (slopeGITT and slopeICI, respectively) above 3.65 V in Figure 4 (Cell1 in cycle 1) plotted against the OCP of the electrode (E) against Li/Li + . The average, 0.00021, is shown by the horizontal line and the standard deviation, 0.076, is shown by the height of the shaded area above and below the average value. Figure 12. Statistical analysis on the relative difference between the diffusion coefficients from the GITT and ICI methods. The relative difference between the Li-ion diffusion coefficient in NMC811 from the GITT and ICI methods (DGITT and DICI, respectively) above 3.7 V in Figure  Supplementary Figure 15. Results from the sequential refinements with a single R3 ) m NMC811 phase and selected XRD patterns near the end of charging and the beginning of discharging. a R-weighted pattern (Rwp) and goodness of fit (GoF) values (top) and c lattice parameters (bottom) from the sequential refinements. Note that, in certain cases, the estimated standard deviation is large due to the poor fit of the model to the data. b Two patterns at the end of charge (pattern 13 and 14) are compared to ones before (pattern 10) and after (pattern 17). It is clearly seen that patterns 13 and 14 are composed of more than one phase. The above has been solved previously 3-6 and is summarized 7,8 as follows for the surface concentration of an electrode particle with a radius rp:

Supplementary
where ƒ is defined as follows: where, αm are the positive roots of α = tan(α), which is listed elsewhere. 5 ƒ(x) has asymptotes in both positive and negative directions. Within 5% error, Equation 3 can be approximated by the following: It can be observed that when x < 0.0032, Equation 3 can be reduced to which is the same as Equation 5 in the main text. The ICI method is derived on the assumption that F(τ1+Δt) ≈ F(τ1) in Equations 14 and 15 in the main text. Suppose a relative error of δ is allowed, the two terms should satisfy the following;

From Equation 3 in this section, it is known that
where, the function and parameters are explained previously. The error can be discussed in terms of both asymptotes of ƒ in Equation 5 since the values are bounded by the asymptotes, which can be found out by plotting the function. Thus, for τ1 < 0.0032rp 2 /D, Let, δ = 0.03, τ1 = 600 s. The maximum Δt is around 37 s. For τ1 > 1.27rp 2 /D,

? 14
It is worth noting that in the standard ICI protocol, the current is only stopped for a few seconds at a time, in contrast to GITT. Therefore, the limit of Δt increases over the number of measurements in the same course of charge or discharge since, τ1 can be effectively accumulated due to the transient current pauses. The limit of Δt discussed above is thus a conservative estimation for the standard ICI protocol.

/ √
With Equation 5 in Supplementary Note 1, the impact of the choice of data selection interval for the GITT analyses can be examined. With r = 2 µm (average particle radius) and D = 10 -11 cm 2 s -1 (which is the maximum value in Figure 5 in the main text), the maximum t is 12.8 s. However, if only potential measurements within 12.8 s are selected, the linear regression renders very large standard deviations due to the limited number of data points. Therefore, the upper limit of the interval is set at 40 s after confirming that the E-√ plots remain linear. The lower limit is set at 5 s since there is a transition region before the linear region on some of the E-√ plots, as reported in previous GITT works on LiNi0.5Mn0.3Co0.2O2 (NMC523). 7 This transition region is prolonged below 3.7 V, as shown in Supplementary Fig. 8a. The phenomenon is also observed in EIS measurements in this work and in the literature for LiNi0.33Mn0.33Co0.33O2 (NMC111), 1,9 where the Warburg element shifts to lower frequencies. Thus, as the cell is discharged below 3.7 V, the linear region on the E-√ plots shifts to 50-150 s. The semi-infinite diffusion assumption still holds here because D is significantly lower here. The situation is more complicated when the cell is charged in the same SoC because the transition region is still long, but D derived from 50-150 s is around 5·10 -11 cm 2 s -1 , which makes 150 s considerably long for the semi-infinite diffusion assumption. Therefore, fitting the data to the full solution in Equations 3 and 4 in Supplementary Note 1 is necessary to derive the diffusion coefficient at SoC below 3.7 V upon charging.

Supplementary Note 3. Analysis of the electrochemical impedance spectra
The impedance spectra above 3.7 V were fitted to the equivalent circuit model (ECM) in Supplementary  Fig. 13. The ECM is adopted from a previous work on NMC111 1 but the finite-space Warburg element is substituted here by a semi-infinite Warburg element because the spectra with the lowest frequency at 10 mHz do not show a vertical tail. The spectra below 3.7 V show enlarged second semicircle (R2 and CPE2) and do not possess sufficient data points to fit the Warburg element, as shown in Supplementary Fig. 10. Some spectra between 3.6 and 3.7 V from Cell 2 can be fitted to the ECM but the majority of the spectra from Cell 2 show a depressed Warburg element with an phase angle of about 22.5°. Therefore, the resulting k and R from the EIS fittings of Cell 2 should be interpreted with caution. Complete set of impedance spectra can be found via Zenodo. 10